# Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics
**Prof. Martin J.
Gander**, Section de Mathématiques, Université de Genève,
will give a public talk (followed by a discussion) on **Wed, July 16, 2008**
at **15:00 o'clock** at **HS 9**
on the topic of
"Schwarz Methods in the course of time"
. The organziers of the symposium,

Univ.-Prof. Dr. Gerhard Larcher

A.Univ.-Prof. Dr. Jürgen Maaß, and

die ÖMG (Österreichische Mathematische Gesellschaft),

hereby cordially invite you.

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

### Schwarz Methods in the course of time

Schwarz domain decomposition methods have been developed at two different levels: at the continuous level for partial differential equations, and at the discrete level of linear systems for parallel computing. At the continuous level, it was Hermann Amandus Schwarz himself who invented the alternating Schwarz method in 1869, as an analytical tool to obtain a rigorous proof of the Dirichlet principle, on which Riemann had founded his theory of analytic functions, without having a proof. More than a century later, in 1989, Pierre-Louis Lions analyzed the method as a tool for parallel computing, and introduced a more parallel variant of the method. At the discrete level, Schwarz methods were formulated by Max Dryja and Olof Widlund in 1987, in the form of Additive Schwarz (AS) and Multiplicative Schwarz (MS). More recently, Restricted Additive Schwarz (RAS) and Additive Schwarz with Harmonic extension (ASH) were discovered by Xiao-Chuan Cai and Marcus Sarkis in 1999, as a result of a programming error.

I will start my talk by showing the historical development of Schwarz methods, pointing out similarities and subtle differences between the classical continuous and discrete Schwarz methods. In particular, I will prove that the more parallel Schwarz method introduced by Pierre-Louis Lions and the Additive Schwarz method are different. I will then introduce at the algebraic level a new class of Schwarz methods, called optimized Schwarz methods, which converge significantly faster than classical Schwarz methods, at the same cost per iteration. These methods are motivated by an idea of Pierre-Louis Lions at the continuous level: they use transmission conditions adapted to the physics of the underlying continuous problem. I will conclude with three important open problems in this area of research.